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APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS
- Publication Year :
- 2020
- Publisher :
- CAMBRIDGE UNIV PRESS, 2020.
-
Abstract
- In this paper, we introduce a method known aspolynomial frame approximationfor approximating smooth, multivariate functions defined on irregular domains in$d$dimensions, where$d$can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the correspondingNikolskii inequalityfor the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.
- Subjects :
- Statistics and Probability
Large class
Multivariate statistics
Polynomial
STOCHASTIC COLLOCATION
POLYNOMIAL-APPROXIMATION
Mathematics, Applied
DIMENSIONALITY
010103 numerical & computational mathematics
NIKOLSKII-TYPE INEQUALITIES
01 natural sciences
Theoretical Computer Science
FOS: Mathematics
Discrete Mathematics and Combinatorics
Mathematics - Numerical Analysis
0101 mathematics
Mathematical Physics
Mathematics
Discrete mathematics
Algebra and Number Theory
Science & Technology
Frame (networking)
ALGORITHMS
Numerical Analysis (math.NA)
ANALYTIC-FUNCTIONS
010101 applied mathematics
Computational Mathematics
Physical Sciences
CHAOS
Geometry and Topology
Hypercube
Analysis
EXTENSION
Subjects
Details
- Language :
- English
- ISSN :
- 20505094
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....90c558d380453303fc9352ea35e003f4